My Arena, My Reign

Xu Da and Zhu Yuanzhang played a game of chess. Assuming Xu Da defeated Zhu Yuanzhang, Xu Da used the chess pieces to form the characters "Long Live" on the chessboard. What is the probability of this event occurring, and why?

The Chinese Mathematical Society's questions really tested the limits of the examinees.

Damn, to solve a math problem, you have to know the rules of Go? I ask you, aren't you angry? Only Chinese students could possibly solve this kind of problem; foreigners would cry after seeing it.

This is the National Final, the most difficult high school math competition in all of China.

If a contestant knows nothing about Go, they're in for a miserable time. How dare you participate in the National Math Final without any special skills?

Shen Qi knows how to play Go, Chinese chess, Ludo, and animal chess. Putting aside his skill level, he understands the rules.

Moreover, Shen Qi knows that many other National Final contestants are familiar with various board game rules and play them quite well. For example, the Nanyue Province Math Olympiad team members can play blindfolded chess.

"Look at the essence through the phenomenon. This question has nothing to do with Zhu Yuanzhang or Xu Da. It's just a math problem. The preceding anecdote is just a smokescreen, except for the last sentence asking for the probability and this chessboard diagram."

Shen Qi quickly thought of Fermat and Pascal's theories on the division of stakes. In a sense, playing chess is also a form of gambling, and there are people who make a living from it under the overpasses.

Since it's gambling, it's indispensable to use the specialized knowledge of Probability Theory and Number Theory.

No matter how the characters "Long Live" were arranged, it was merely a probabilistic event, a little trick for those who understand mathematics. If Zhu Yuanzhang understood math, he would have immediately punished Xu Da instead of rewarding him with Mochou Lake.

The theory of the division of stakes, jointly drafted by Fermat and Pascal, and the subsequent derivative theories are the theoretical basis for the world's major casinos to consistently make profits. Calculating the probability of forming the characters "Long Live" is similar to the theoretical principle of calculating the probability of getting two aces and four twos and being left with a straight.

Shen Qi began to write: Let black pieces be p, and white pieces be q. If p is the probability of an event occurring once, then q is the probability of the event not occurring.

Then, the probability of the event occurring at least m times in n trials is equal to the sum of terms from the p^n term to the term with p^m * q^(n-m) in the expansion of (p+q)^n.

Based on this theory, Shen Qi quickly calculated the probability of the characters "Long Live" appearing as a mere 0.0002, and elaborated on the reasons.

Calculating probability isn't difficult. If you master the mathematical principles above, you can become a gambling king. What's difficult is surviving the casino with all your limbs intact.

Shen Qi inferred that Zhu Yuanzhang and Xu Da did play chess, but the story of Xu Da forming "Long Live" to win Mochou Lake was likely just a legend.

From a mathematical perspective, you'd have to play fifty thousand games of chess to see "Long Live" appear once. A game of chess can be short, taking tens of minutes, or long, taking several hours. Playing three to five games a day is considered a lot.

If Zhu Yuanzhang and Xu Da did nothing but play chess every day, it would take them 27 years to see "Long Live" formed by black and white pieces.

Zhu Yuanzhang was the founding emperor; didn't he have state affairs to attend to?

Of course, it's also possible that the "Long Live" event randomly appeared in the first of fifty thousand times.

So, this is just a legend and shouldn't be taken seriously.

"This first question, it seemed painful at first glance, but after solving it, the pain in my testicles disappeared, and I even felt a slight thrill. This question is actually quite interesting. Ah, I've never been to Mochou Lake, I really want to see it." Shen Qi ate a Snickers bar to celebrate his successful solution of the first question in the National Final.

Without pausing, Shen Qi moved on to the second question, which was a Plane Analytic Geometry problem.

For Shen Qi, who was at Level 5 in mathematics, affine transformations in two-dimensional homogeneous coordinates were not difficult to analyze using determinants. It was simply a matter of finding a set of invariants for rotation, translation, and reflection.

The singly-periodic elliptic geometry corresponds to a subgroup of projective transformations. This seems like an obvious axiom, but one must not be misled by its appearance, or they will go astray.

The most rational Math Olympiad contestants only need to cut to the chase and find the absolute conic on the plane. The second question was a freebie.

Shen Qi used an economical and practical method to find the absolute conic. The Klein's continuous transformation described in university textbooks was too complicated and simply created unnecessary trouble.

Poincare, hailed as the last "universal scholar" in the world, was clearly more flexible. Shen Qi often applied many of Poincare's viewpoints and conclusions.

In a plane coordinate system, there are many ways to obtain the absolute conic from a curve. Poincare's degenerate superposition method is practically a divine tool for competition formats. Shen Qi used this degenerate superposition method, which was direct, simple, and brutal. It was very effective, as if tailor-made for math competitions.

Two hours passed, and Shen Qi had solved two problems. He took a sip of Dongpeng Special Drink to energize himself, accompanied by a bear cookie, a Snickers bar, and a Wife Cake to replenish his energy.

Don't think of mathematics as purely a mental activity; it's also physically demanding. Just sitting there without moving and writing for two consecutive hours will tire you out.

Sixty National Final contestants were distributed across seven classrooms. Shen Qi's classroom had ten contestants, each from a different province or city.

There were as many as eleven invigilators, with one-on-one supervision leaving one to roam freely.

After all, this is the National Final. A gold medal in the Math Olympiad is extremely valuable. After a rigorous selection process, only six individuals will receive a gold medal. These six lucky winners will undoubtedly become highly sought-after by major universities.

The invigilator responsible for monitoring Shen Qi walked behind him and softly hummed, "Are you full, student?"

"Yes, I'm full. I'll have another spicy strip for good luck, wishing for prosperity." Shen Qi, with a spicy strip in his left hand and a compass in his right, drew concentric circles while eating.

"Student, why did you bring so many snacks into the exam hall?" the invigilator asked in a low voice, frowning.

Shen Qi found it strange, "Huh? Didn't you allow food? It's 4.5 hours, teacher, almost like the ancient imperial examinations. It's a battle of endurance; you'll get hungry if you don't eat."

"Eat less; eating too much can make you drowsy," the invigilator kindly reminded him. He glanced at Shen Qi's exam paper. Nanyue Province Math Olympiad team, Shen Qi. Not bad, this young boy finished two problems in just two hours and is leisurely eating spicy strips. He might be a dark horse!

The invigilator monitoring Shen Qi was a staff member of the Chinese Mathematical Society and a professional mathematics researcher himself. He paced to the classroom door and asked a younger man, "National Final contestant number 56, Shen Qi, what was his score in the preliminary round?"

The young man, holding an iPad, pulled up the information and said, "Shen Qi, preliminary round score: 21 points, perfect score. It's not easy for a Nanyue team contestant who barely qualified for the National Final to get a perfect score. Shen Qi is the only one on their team with a perfect preliminary score."

"Hmm, he's talented. Our society's principle is to recruit talent without regard to background," the older invigilator nodded and said to the young man, "Xiao Wen, go buy me a pack of spicy strips, the same kind Shen Qi brought. It's past 11 o'clock, I'm hungry."

(Hehehe, students, please give some recommendation votes (▽))